Differential Equations-Exercise 12.1

1.Please see the question from book.

Soln:

Or, dydx= xy

Or, x.dx = y.dy

Integrating, x22=y22+c2

So, x2 = y2 + c.

Or, x2 – y2 = c.

2.Please see the question from book.

Soln:

Or, x.dx + y.dy = 0

Integrating, x22+y22=c2

So, x2 + y2 = c.

3.Please see the question from book.

Soln:

Or, dydx$\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}$ = x2+1y2+1à (y2 + 1).dy = (x2 + 1).dx

Integrating, y33+ y + c = x33 + x.

4.Please see the question from book.

Soln:

Or, x2.dy – y2.dx = 0

Dividing each term by x2y2.

Or, dyy2−dxx2= 0.

Integrating, −1y−(−1x) = c’

Or, −1y+1x = c’

Or, - x + y = c’xy.

So, x – y = c’xy

So, x – y = cxy, where c = c’

5.Please see the question from book.

Soln:

Or, (1 + x)2dydx= 1.

Or, dy =dx1+x2

Integrating, y = tan-1 x + c.

6.Please see the question from book.

Soln:

Or, dydx= ex+1y

Or, y.dy = (ex + 1).dx

Integrating, y22 = ex + x + c’

So, y2 = 2ex + 2x + 2c’

So, y2 = 2ex + 2x + c where, c = 2c’

7.Please see the question from book.

Soln:

Or, dydx + 4x = 2e2x

Or, dy + 4x.dx = 2e2x.dx

Integrating, y + 2x2 = 2.e2x2 + C.

So, y= e2x – 2x2 + C.

8.Please see the question from book.

Soln:

Or, 1−x2−−−−−√.dy + 1−y2−−−−−√.dx = 0.

Dividing each term by 1−x2−−−−−√.1−y2−−−−−√

Or, dy1−y2√+ dx1−x2√= 0.

Integrating, sin-1y + sin-1x = sin-1c

Or, sin-1{x1−y2−−−−−√+y1−x2−−−−−√} = sin-1c

So, x1−y2−−−−−√ + y.1−x2−−−−−√= c.

9.Please see the question from book.

Soln:

Or, (1 + x)y.dx + (1 + y)x.dy = 0.

Dividing each term by xy,

Or, (1+xx).dx + (1+yy)dy 0

Or, (1x+1).dx + (1y+1).$.$dy = 0

Integrating,

Logx + x + log y + y = c.

Or, log(xy) + x + y = c.

10.Please see the question from book.

Soln:

Or, (xy2 + x).dx + (yx2 + y).dy = 0

Or, x(y2 + 1).dx + y(x2 + 1).dy = 0

Dividing each term by (x2 + 1)(y2 + 1),

Or, xx2+1$\frac{+1}{}$.dx + yy2+1$\frac{+1}{}$.dy = 0.

Integrating, 12log(x2 + 1) + 12log(y2 + 1) = 12.logc.

Or, log(x2 + 1)(y2 + 1) = logc.

Or, (x2 + 1)(y2 + 1) = c.

11.Please see the question from book.

Soln:
or, x.dydx + y – 1 = 0.

Or, x.dy + y.dx – dx = 0.

Or, d(xy) – dx = 0.

Integrating, xy – x = C àx(y – 1) = C.

12.Please see the question from book.

Soln:

Or, dydx = ex – y  + x3.e-y.

Or, ey.dy = (ex + x3).dx 

Integrating, ey = ex + x44 + C.

13.Please see the question from book.

Soln:

Or, tanx.dy + tany.dx = 0

Dividing each term by tanx.tany

Or, dytany + dxtanx = 0.

Or, cosysiny.dy + cosxsinx.dx = 0

Integrating,

Or, log(siny) + log(sinx) = logc.

Or, log(sinx.siny) = logc.

So, sinx.siny = c.

14.Please see the question from book.

Soln:

Or, dydx= −1+cos2y1−cos2x

Or, dydx$\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}$ = −2cos2y2sin2x

Or, −dxsin2x = dycos2y

Or, -cosec2x.dx = sec2y.dy

Integrating, cotx = tany + c.

So, cotx – tany = c.

If you find any mistakes in these above solutions please let me know in the comments i will greatly appreciate that. and try to correct that as well.